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Period Integrals and the Riemann-Hilbert Correspondence

Huang, An and Lian, Bong H. and Zhu, Xinwen (2016) Period Integrals and the Riemann-Hilbert Correspondence. Journal of Differential Geometry, 104 (2). pp. 325-369. ISSN 0022-040X.

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A tautological system, introduced in [20][21], arises as a regular holonomic system of partial differential equations that governs the period integrals of a family of complete intersections in a complex manifold X, equipped with a suitable Lie group action. A geometric formula for the holonomic rank of such a system was conjectured in [5], and was verified for the case of projective homogeneous space under an assumption. In this paper, we prove this conjecture in full generality. By means of the Riemann–Hilbert correspondence and Fourier transforms, we also generalize the rank formula to an arbitrary projective manifold with a group action.

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Additional Information:© 2016 International Press. Received June 24, 2015. S. Bloch has independently noticed the essential role of the Riemann–Hilbert correspondence in connecting the de Rham cohomology and solution sheaf of a tautological system. We thank him for kindly sharing his observation with us. We also thank T. Lam for helpful communications. A.H. would like to thank S.-T. Yau for advice and continuing support, especially for providing valuable resources to facilitate his research. B.H.L. is partially supported by NSF FRG grant DMS 1159049. X.Z. is supported by NSF grant DMS-1313894 and DMS-1303296 and the AMS Centennial Fellowship.
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Record Number:CaltechAUTHORS:20140919-164509616
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:49879
Deposited By: George Porter
Deposited On:22 Sep 2014 20:47
Last Modified:03 Oct 2019 07:18

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